Multiplying Algebraic Terms
What is $3x \times 4y$? Multiply the numbers, multiply the variables. Learn the simple rules that make algebraic multiplication effortless.
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What is $3a \times 4b$? Explain your reasoning about how you multiply the numbers and the letters separately.
To multiply algebraic terms, multiply the coefficients (the numbers) and multiply the variables (the letters) separately. Then write them together.
$3x \times 4y$: coefficients $3 \times 4 = 12$, variables $x \times y = xy$. Result: $12xy$. Unlike addition, you CAN multiply unlike terms!
Know
- Coefficients multiply together
- Variables multiply together
- Same variable = add powers
Understand
- Why $x \times x = x^2$
- Why order of letters doesn't matter
- How to handle negatives
Can Do
- Multiply two or more algebraic terms
- Simplify products with same variables
- Handle multiple variables
Wrong: $x \times x = 2x$
Right: $x \times x = x^2$. It's $x$ multiplied by $x$, not $x$ plus $x$.
Wrong: $3x \times 2x = 6x$
Right: $3x \times 2x = 6x^2$. Coeff: $3 \times 2 = 6$. Var: $x \times x = x^2$.
When you multiply terms with the same variable, you add the powers. $x^2 \times x^3 = x^{2+3} = x^5$.
$x^2 = x \times x$ and $x^3 = x \times x \times x$. So $x^2 \times x^3 = (x \times x)(x \times x \times x) = x \times x \times x \times x \times x =$ $x^5$.
| Product | Coefficients | Variables | Result |
|---|---|---|---|
| $2x \times 3x$ | $2 \times 3 = 6$ | $x \times x = x^2$ | $6x^2$ |
| $3a \times 4ab$ | $3 \times 4 = 12$ | $a \times ab = a^2b$ | $12a^2b$ |
| $x^2 \times x^3$ | $1 \times 1 = 1$ | $x^{2+3} = x^5$ | $x^5$ |
| $5xy \times 2y$ | $5 \times 2 = 10$ | $xy \times y = xy^2$ | $10xy^2$ |
Multiply term by term. Gather same variables. Write in alphabetical order.
Use the sign rules: positive × positive = positive, negative × negative = positive, positive × negative = negative.
| Product | Working | Answer |
|---|---|---|
| $3x \times 4y$ | $3 \times 4 = 12$, $xy$ | $12xy$ |
| $2a \times 5a$ | $2 \times 5 = 10$, $a^2$ | $10a^2$ |
| $(-3x)(2y)$ | $-3 \times 2 = -6$, $xy$ | $-6xy$ |
| $x^2 \times x^4$ | $1$, $x^{2+4}$ | $x^6$ |
| $4xy \times 3x$ | $4 \times 3 = 12$, $x^2y$ | $12x^2y$ |
Watch Me Solve It · Worked example
- 1Multiply the coefficients$(-2) \times 3 \times (-1) = +6$$-2 \times 3 = -6$. Then $-6 \times (-1) = +6$. Two negatives make a positive!
- 2Multiply the $x$ variables$x \times x = x^2$The first term has $x^1$ (just $x$), the second has $x^1$. $x^1 \times x^1 = x^{1+1} = x^2$.
- 3Multiply the $y$ variables$y \times y = y^2$The second term has $y^1$, the third has $y^1$. $y^1 \times y^1 = y^2$.
- 4Combine all parts$6x^2y^2$Coefficient: 6. $x$-part: $x^2$. $y$-part: $y^2$. Write in alphabetical order: $x$ before $y$.
How are you completing this lesson?
Brain Trainer · 4 problems
1 Simplify $4x \times 3y$.
$4 \times 3 = 12$, $x \times y = xy$. $12xy$$12xy$2 Simplify $2a \times 5a$.
$2 \times 5 = 10$, $a \times a = a^2$. $10a^2$$10a^2$3 Simplify $(-3x)(-2y)$.
$-3 \times -2 = +6$. $6xy$$6xy$4 Simplify $x^2 \times x^3 \times 2x$.
Coeff: $1 \times 1 \times 2 = 2$. $x$-part: $x^{2+3+1} = x^6$. $2x^6$$2x^6$
Show Your Working · 3 questions
Q6. Simplify $5a \times 2b$.
Q7. Simplify $(-3x)(2y)(-x)$.
Q8. A box has dimensions $2x$, $3x$, and $y$. Write and simplify an expression for its volume.
Stretch. If $2x \times 3x \times k = 30x^2$, find the value of $k$.
Extension Problems
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Key Concept
Review the main ideas from this lesson.
Formulas
Key formulas and rules.
Watch Out
Common mistakes to avoid.
Check
Always verify your answers.
Practice
Keep practicing to master.
Next
Build on these skills.
Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
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